On the hessian of the optimal transport potential

Stefán Ingi Valdimarsson

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 3, page 441-456
  • ISSN: 0391-173X

Abstract

top
We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove Brascamp-Lieb inequalities and thus prove a new generalised Reverse Brascamp-Lieb inequality.

How to cite

top

Valdimarsson, Stefán Ingi. "On the hessian of the optimal transport potential." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.3 (2007): 441-456. <http://eudml.org/doc/272250>.

@article{Valdimarsson2007,
abstract = {We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove Brascamp-Lieb inequalities and thus prove a new generalised Reverse Brascamp-Lieb inequality.},
author = {Valdimarsson, Stefán Ingi},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Monge-Kantorovich mass transport problem; bounds for the Hessian of the optimal transport potential; Brascamp-Lieb inequalities},
language = {eng},
number = {3},
pages = {441-456},
publisher = {Scuola Normale Superiore, Pisa},
title = {On the hessian of the optimal transport potential},
url = {http://eudml.org/doc/272250},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Valdimarsson, Stefán Ingi
TI - On the hessian of the optimal transport potential
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 3
SP - 441
EP - 456
AB - We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove Brascamp-Lieb inequalities and thus prove a new generalised Reverse Brascamp-Lieb inequality.
LA - eng
KW - Monge-Kantorovich mass transport problem; bounds for the Hessian of the optimal transport potential; Brascamp-Lieb inequalities
UR - http://eudml.org/doc/272250
ER -

References

top
  1. [1] S. Alesker, S. Dar, and V. Milman, A remarkable measure preserving diffeomorphism between two convex bodies in ℝ n , Geom. Dedicata74 (1999), 201–212. Zbl0927.52007MR1674116
  2. [2] F. Barthe, On a reverse form of the Brascamp–Lieb inequality, Invent. Math.134 (1998), 335–361. Zbl0901.26010MR1650312
  3. [3] J. Bennett, A. Carbery, M. Christ and T. Tao, The Brascamp–Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal., to appear. Zbl1132.26006MR2377493
  4. [4] Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math.305 (1987), 805–808. Zbl0652.26017MR923203
  5. [5] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math.44 (1991), 375–417. Zbl0738.46011MR1100809
  6. [6] L. A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys.214 (2000), 547–563. Zbl0978.60107MR1800860
  7. [7] L. A. Caffarelli, Erratum: Monotonicity of optimal transportation and the FKG and related inequalities, Comm. Math. Phys. 214 (2000), 547–563; Comm. Math. Phys. 225 (2002), 449–450. Zbl0978.60107
  8. [8] W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math.177 (1996), 113–161. Zbl0887.49017MR1440931
  9. [9] R. A. Horn and C. R. Johnson, “Matrix Analysis”, Cambridge University Press, Cambridge, 1985. Zbl1267.15001MR832183
  10. [10] E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math.102 (1990), 179–208. Zbl0726.42005MR1069246
  11. [11] C. Villani, “Topics in Optimal Transportation”, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003. Zbl1106.90001MR1964483
  12. [12] C. Villani, “Optimal Transport, Old and New”, 2007, July 18, preprint. Zbl1156.53003MR2459454

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.